Twists of elliptic curves with CM

Abstract: We consider certain families of sextic twists $E_D$ of the elliptic curve $y^2=x^3+1$ that are not defined over $\mathbb{Q}$, but over $\mathbb{Q}[\sqrt{-3}]$.

We compute a formula that relates the $L$-value $L(E_D, 1)$ to the square of a trace of a modular function at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture, when the value above is non-zero, we should recover the order of the Tate-Shafarevich group, and under certain conditions we show that the value is indeed a square.

algebraic geometrynumber theory

Audience: researchers in the topic

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Rendez-vous on special values and periods

Series comments: The main objective of this conference is to gather together young researchers interested in special values of L-functions and periods. These objects are at the crossroads of many recent important developments in arithmetic geometry, such as Euler systems or the theory of motives. The different talks will portray the variety of viewpoints with which L-functions and periods are studied at present.

Registration is free and mandatory, to get access to the livestream and recording of the talks.

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